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Mechanics 2/23 Mechanics 2/23 Lecture notes, 2010/11, teaching block 1 Contents 0 Introduction 1 1 The calculus of variations 5 1.1 Example.................................. 5 1.2 Generalisation............................... 6 1.3 Euler-Lagrangeequation. 8 1.4 Solutionofourproblem .. .. .. .. .. .. .. 10 1.5 Alternative version of the Euler-Lagrange equation . ....... 10 1.6 Thebrachistochrone ........................... 11 1.7 Functionals depending on several functions . 16 1.8 Fermat’sprinciple ............................ 16 2 Lagrangian mechanics 21 2.1 Reminder:Newton ............................ 21 2.2 Lagrangian mechanics in Cartesian coordinates . ..... 22 2.3 Generalised coordinates and constraints . 24 2.3.1 Gravitational field . 26 2.3.2 Pendulum............................. 27 2.3.3 Inclinedplane........................... 28 2.3.4 General properties of forces of constraint . 33 2.3.5 Derivation of Lagrange’s equations from Newton’s law in the generalcase............................ 35 2.4 Conservedquantities . .. .. .. .. .. .. .. 39 2.4.1 Energy conservation . 39 2.4.2 Conservation of generalised momenta . 45 2.4.3 Sphericalpendulum . 47 2.4.4 Noether’stheorem . .. .. .. .. .. .. 55 3 Small oscillations 63 3.1 Generaltheory .............................. 63 3.2 Twosprings................................ 67 3.3 Small oscillations about equilibrium . 70 3.4 Thedoublependulum .......................... 73 4 Rigid bodies 79 4.1 Angularvelocity ............................. 80 4.2 Inertiatensor ............................... 81 4.3 Euler’sequations ............................. 86 i ii CONTENTS 5 Hamiltonian mechanics 91 5.1 Hamilton’sequations. .. .. .. .. .. .. .. .. 91 5.2 Conservation laws and Poisson brackets . 98 5.3 Canonicaltransformations . 103 Chapter 0 Introduction In this course we are going to consider a formulation of mechanics (variational mechanics) that is different from the one in Mechanics 1 (Newtonian mechanics). I first want to explain why such a formulation is called for, and discuss some of the main ideas. Newtonian mechanics The central result in Newtonian mechanics concerns the acceleration of a particle, i.e., the second derivative of its position r w.r.t. time. This acceleration is given by the ratio of the force F acting on the particle and its mass m, i.e., d2r F = . dt2 m The force F typically depends on the particle position r; in addition it may depend on the positions of other particles and on the velocities of the particles involved. Thus we obtain differential equations that contains both particle positions and their second (and sometimes its first) derivatives, i.e., second-order ordinary differ- ential equations. These differential equations provide a way of determining the trajectories of par- ticles. However, their direct application can sometimes become messy. Historically, the first applications where this was felt in engineering and in astronomy. Variational mechanics Figure 1: Isaac Newton, Joseph Louis Lagrange, and William Rowan Hamilton. To simplify the treatment of such complex problems, a variational formulation of mechanics was developped. There are two different versions of variational me- 1 2 CHAPTER 0. INTRODUCTION chanics, respectively introduced by Joseph Louis Lagrange (1736-1813) and William Rowan Hamilton (1805-1865). To explain the key idea of these approaches, let us consider the simplest case possible: one particle on which no forces are acting. d2r dr We then have dt2 = 0, i.e., the particle travels with constant velocity dt along a straight line. Now the important point is that the straight line is also the short- est connection between two points. Thus, instead of using a differential equation, we would have obtained a trajectory of the same form if we had postulated that without forces particles always follow the shortest line connecting two points. The main idea of variational mechanics is to generalise this simple observation to more general settings. Our goal is thus to formulate mechanics through a varia- tional principle: Particles travel on trajectories that extremise (minimise, maximise) “something”. This “something” is not always the length, and one of our goals will be to find what it is. In general it will be called the “action”. Advantages of variational mechanics The main advantages of the variational approach to mechanics are: In mechanics it is often helpful to work in coordinates adapted to the system • we are interested in. For example, if we want to describe the motion of a satellite in the gravitational field of the Earth, it is helpful to use spherical coordinates with the centre of the Earth as the origin. More complicated systems require other complicated sets of coordinates. For example if we consider a particle moving in the gravitational field of two masses it would be better to use elliptic coordinates (see Fig. 2). In these coordinates two points are singled out, corresponding to the positions of the two masses. Figure 2: Elliptic coordinates. We shall see that in variational mechanics it becomes particularly simple to switch between coordinate systems. 3 Many mechanical systems have constraints, i.e., conditions where particles or • bodies are allowed to be and where not. A practical example would be a train that is required to stay on the railroad track. In variational mechanics these constraints can easily be incorporated by choosing appropriate coordinates, e.g., by taking the railroad track as a coordinate line. Crucially, many areas of mathematical physics are formulated in the lan- • guage developped in variational mechanics. For instance “Hamiltonians” and “Lagrangians” play an important role in quantum mechanics or chaos the- ory. Similarly the ideas underlying variational mechanics are important in the theory of differential equations. (Rough) outline of this course In this course, we will first develop the mathematical tools needed for extremisation problems like the one sketched above (variational calculus). We will then consider both the Lagrangian and the Hamiltonian formulation of mechanics and several examples for their use. Reminder: Chain rule As a preparation for many calculations in this course I want to remind how deriva- tives of expressions like f(u(t), v(t),t) are evaluated. According to the chain rule, we first have to differentiate w.r.t. the arguments of f, and then multiply with the derivatives of these arguments w.r.t. t. This yields d ∂f du ∂f dv ∂f f(u(t), v(t),t)= (u(t), v(t),t) + (u(t), v(t),t) + (u(t), v(t),t) . dt ∂u dt ∂v dt ∂t du dv Dropping the arguments and settingu ˙ = dt , v˙ = dt , we can also write df ∂f ∂f ∂f = u˙ + v˙ + . dt ∂u ∂v ∂t ∂f It is important to distinguish between the partial derivative ∂t with respect to ∂f the third argument of f, and the total derivative ∂t where also the t-dependence of the first and the second argument are taken into account. An example would be f representing the termperature felt by a person, f de- noting the time, and u(t), v(t) the position of the person. Then the rate of change df ∂f dt of the temperature felt will depend on the change of temperature with time ( ∂t ) and on whether the person moves to a warmer or colder place (leading to the terms ∂f ∂f ∂u u˙ and ∂v v˙). 4 CHAPTER 0. INTRODUCTION Chapter 1 The calculus of variations 1.1 Example Figure 1.1: Possible connections between (x1,y1) and (x2,y2). To introduce variational mechanics, we first need to equip ourselves with the tools needed to solve extremisation problems. We will start with the simplest ex- ample: Showing that the shortest connection between two points in R2 is a straight line. Let us denote these two points by coordinates (x1,y1) and (x2,y2) in a Cartesian coordinate system, see Fig. 1.1. Curves connecting these two points can then be described through functions y(x) that assign to each x-coordinate between x1 and x2 the corresponding y-coordinate. We now have to determine the length l of the curve corresponding to each function y(x) • and find a y(x) such that l becomes minimal. • To start with the first task, we split the x-axis into small pieces of length dx. As seen in Fig. 1.2, this means that the curve is also split into small pieces, whose length will be denoted by dl. We assume that the pieces are small enough such that inside each piece the curve can be considered as a straight line. This means that for each piece we can draw a triangle as in Fig. 1.2 with the width dx, the height 5 6 CHAPTER 1. THE CALCULUS OF VARIATIONS Figure 1.2: Determining the length of a curve y(x) in R2. dy, and the length dl as its sides. Since the slope of the curve must be given by the derivative y′(x), dy depends on dx as dy y′(x)= dy = y′(x)dx . dx ⇒ We can now determine the length of each piece using Pythagoras’ theorem, 2 2 2 2 2 (dl) = (dx) + (dy) = (1+ y′(x) )(dx) dl = 1+ y (x)2dx . ⇒ ′ To obtain the overall length of the curve, we have to sum over the lengths dl of all pieces. If we take the limit where the width dx of the pieces goes to zero, this sum can be replaced by the integral, and we obtain the Length of a curve in R2 x2 2 l = 1+ y′(x) dx . x1 Our task is now to find functions y(x) with y(x1) = y1 and y(x2) = y2 for which the length l becomes minimal. 1.2 Generalisation The problem outlined above is the simplest example for a more general type of variational problems, where we maximise, minimise, or in general look for stationary points of an integral of a function depending on y(x), y′(x) and x: 1.2. GENERALISATION 7 Stationarity problem Let x2 K[y]= f(y(x),y′(x),x)dx .
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